There has been recent notable work that aims to characterize the epidemic thresholds of these compartmental models with disease transmission over a network [1]}, [2]}. The critical idea is that the epidemic threshold of a network can be characterized as the inverse of the greatest (first) eigenvalue of the adjacency matrix \(A\)  of the network, denoted as:
\(\boxed{\tau (A) = \dfrac{1}{\lambda _1(A)}}\) 
